Key Concepts and Formulas

• Complex Number Representation: A complex number $z$ can be represented as $z = x + iy$, where $x$ and $y$ are real numbers, and $i = \sqrt{-1}$.
• Imaginary Part: For a complex number $z = x + iy$, the imaginary part is denoted as $Im(z) = y$.
• Equation of a Circle: The general equation of a circle in the Cartesian plane is $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $R = \sqrt{g^2 + f^2 - c}$.

Step-by-step Solution

Given the equation: $Im\left( {{{iz - 2} \over {z - i}}} \right) + 1 = 0$ which implies $Im\left( {{{iz - 2} \over {z - i}}} \right) = -1$

Step 1: Substitute $z = x + iy$

We begin by substituting $z = x + iy$ into the complex expression. Why this step? To convert the complex number equation into an equation involving real variables $x$ and $y$, which is necessary to determine its geometric representation.

The numerator becomes: $iz - 2 = i(x + iy) - 2 = ix + i^2y - 2 = ix - y - 2 = (-y - 2) + ix$

The denominator becomes: $z - i = (x + iy) - i = x + i(y - 1)$

So the expression is: ${{(-y - 2) + ix} \over {x + i(y - 1)}}$

Step 2: Rationalize the complex fraction

To find the imaginary part of a complex fraction $\frac{A + iB}{C + iD}$, we multiply both the numerator and the denominator by the conjugate of the denominator, which is $C - iD$. Why this step? This process eliminates the imaginary term from the denominator, transforming it into a real number. This makes it straightforward to separate the real and imaginary parts of the entire fraction.

The conjugate of the denominator $x + i(y - 1)$ is $x - i(y - 1)$. \begin{align*} {{{(-y - 2) + ix} \over {x + i(y - 1)}}} &= {{{(-y - 2) + ix} \over {x + i(y - 1)}} \times {{x - i(y - 1)} \over {x - i(y - 1)}}} \\ &= {{[(-y - 2) + ix][x - i(y - 1)]} \over {x^2 - (i(y - 1))^2}} \\ &= {{[(-y - 2) + ix][x - i(y - 1)]} \over {x^2 - i^2(y - 1)^2}} \\ &= {{[(-y - 2) + ix][x - i(y - 1)]} \over {x^2 + (y - 1)^2}} \quad (\text{since } i^2 = -1) \end{align*}

Step 3: Expand the numerator and group real and imaginary parts

Now we expand the numerator: \begin{align*} {[(-y - 2) + ix][x - i(y - 1)]} &= (-y - 2)x + (-y - 2)(-i(y - 1)) + ix(x) + ix(-i(y - 1)) \\ &= -xy - 2x + i(y + 2)(y - 1) + ix^2 - i^2x(y - 1) \\ &= -xy - 2x + i(y^2 + y - 2) + ix^2 + x(y - 1) \\ &= -xy - 2x + i(y^2 + y - 2) + ix^2 + xy - x \\ \end{align*} Group the real and imaginary terms in the numerator: Real part: $(-xy - 2x + xy - x) = -3x$ Imaginary part: $i(y^2 + y - 2 + x^2)$

So the full expression becomes: ${{-3x + i(x^2 + y^2 + y - 2)} \over {x^2 + (y - 1)^2}}$

Step 4: Equate the imaginary part to -1

From the problem statement, $Im\left( {{{iz - 2} \over {z - i}}} \right) = -1$. The imaginary part of the simplified expression is $\frac{x^2 + y^2 + y - 2}{x^2 + (y - 1)^2}$. Why this step? This is the direct application of the given condition to form the equation of the locus.

Therefore, we have: ${{x^2 + y^2 + y - 2} \over {x^2 + (y - 1)^2}} = -1$

Step 5: Simplify the equation to the standard form of a circle

Multiply both sides by the denominator (note that $z \neq i$, so $x^2 + (y - 1)^2 \neq 0$): $x^2 + y^2 + y - 2 = -[x^2 + (y - 1)^2]$ $x^2 + y^2 + y - 2 = -x^2 - (y^2 - 2y + 1)$ $x^2 + y^2 + y - 2 = -x^2 - y^2 + 2y - 1$

Now, bring all terms to one side: $x^2 + y^2 + y - 2 + x^2 + y^2 - 2y + 1 = 0$ $2x^2 + 2y^2 - y - 1 = 0$

Divide by 2 to get the standard form $x^2 + y^2 + 2gx + 2fy + c = 0$: $x^2 + y^2 - {1 \over 2}y - {1 \over 2} = 0$

Step 6: Identify the center and calculate the radius

For a circle $x^2 + y^2 + 2gx + 2fy + c = 0$: Here, $2g = 0 \Rightarrow g = 0$ $2f = -{1 \over 2} \Rightarrow f = -{1 \over 4}$ $c = -{1 \over 2}$

The radius $R$ is given by the formula $R = \sqrt{g^2 + f^2 - c}$. Why this step? Once the equation is in standard form, we can directly apply the radius formula.

\begin{align*} R &= \sqrt{(0)^2 + \left(-{1 \over 4}\right)^2 - \left(-{1 \over 2}\right)} \\ &= \sqrt{0 + {1 \over {16}} + {1 \over 2}} \\ &= \sqrt{{1 \over {16}} + {8 \over {16}}} \\ &= \sqrt{{9 \over {16}}} \\ &= {3 \over 4} \end{align*}

The radius of the circle is $3/4$.

Common Mistakes & Tips

• Sign Errors: Pay close attention to signs when expanding and simplifying expressions, especially when dealing with $i^2 = -1$.
• Complex Conjugate: Double-check the multiplication by the complex conjugate to ensure correct expansion and separation of real and imaginary parts.
• Standard Equation of Circle: Make sure the coefficients of $x^2$ and $y^2$ are both 1 before applying the radius formula.

Summary

By substituting $z = x + iy$ into the given equation and carefully simplifying the complex expression, we obtained the Cartesian equation of a circle. After identifying the coefficients in the standard equation, we calculated the radius of the circle to be $\frac{3}{4}$.

The final answer is $\boxed{{3 \over 4}}$, which corresponds to option (C).